The Russian Premier League club have been linked with a move for the Rennes star after he returned from the recent African Cup of Nations with the player-of-the-tournament award.
Pitroipa, 26, is in his second season at the French club after a disappointing spell at Bundesliga side SV Hamburg and could follow in the footsteps of his former Rennes team-mate Yann M'Vila by moving to Russia.
Rubin beat Fulham and QPR to the signature of French international M'Vila in January, paying £9.6million for the 22-year-old midfielder.
However, he is now being investigated by French police for allegedly being involved with an under-age prostitute.
Rubin could potentially outbid Fulham again for Pitroipa, who has a contract until June 2015, and the Russians may step up their pursuit after Zenit St Petersburg were linked to their Finnish winger Roman Eremenko.